| Preface | p. ix |
| Elementary operator theory | p. 1 |
| Banach spaces | p. 1 |
| Bounded linear operators | p. 12 |
| Topologies on vector spaces | p. 19 |
| Differentiation of vector-valued functions | p. 23 |
| The holomorphic functional calculus | p. 27 |
| Function spaces | p. 35 |
| L[superscript p] spaces | p. 35 |
| Operators acting on L[superscript p] spaces | p. 45 |
| Approximation and regularization | p. 54 |
| Absolutely convergent Fourier series | p. 60 |
| Fourier transforms and bases | p. 67 |
| The Fourier transform | p. 67 |
| Sobolev spaces | p. 77 |
| Bases of Banach spaces | p. 80 |
| Unconditional bases | p. 90 |
| Intermediate operator theory | p. 99 |
| The spectral radius | p. 99 |
| Compact linear operators | p. 102 |
| Fredholm operators | p. 116 |
| Finding the essential spectrum | p. 124 |
| Operators on Hilbert space | p. 135 |
| Bounded operators | p. 135 |
| Polar decompositions | p. 137 |
| Orthogonal projections | p. 140 |
| The spectral theorem | p. 143 |
| Hilbert-Schmidt operators | p. 151 |
| Trace class operators | p. 153 |
| The compactness of f(Q)g(P) | p. 160 |
| One-parameter semigroups | p. 163 |
| Basic properties of semigroups | p. 153 |
| Other continuity conditions | p. 577 |
| Some standard examples | p. 182 |
| Special classes of semigroup | p. 190 |
| Norm continuity | p. 190 |
| Trace class semigroups | p. 194 |
| Semigroups on dual spaces | p. 197 |
| Differentiable and analytic vectors | p. 201 |
| Subordinated semigroups | p. 205 |
| Resolvents and generators | p. 210 |
| Elementary properties of resolvents | p. 210 |
| Resolvents and semigroups | p. 218 |
| Classification of generators | p. 227 |
| Bounded holomorphic semigroups | p. 237 |
| Quantitative bounds on operators | p. 245 |
| Pseudospectra | p. 245 |
| Generalized spectra and pseudospectra | p. 251 |
| The numerical range | p. 264 |
| Higher order hulls and ranges | p. 276 |
| Von Neumann's theorem | p. 285 |
| Peripheral point spectrum | p. 287 |
| Quantitative bounds on semigroups | p. 296 |
| Long time growth bounds | p. 296 |
| Short time growth bounds | p. 300 |
| Contractions and dilations | p. 307 |
| The Cayley transform | p. 310 |
| One-parameter groups | p. 315 |
| Resolvent bounds in Hilbert space | p. 321 |
| Perturbation theory | p. 325 |
| Perturbations of unbounded operators | p. 325 |
| Relatively compact perturbations | p. 330 |
| Constant coefficient differential operators on the half-line | p. 335 |
| Perturbations: semigroup based methods | p. 339 |
| Perturbations: resolvent based methods | p. 350 |
| Markov chains and graphs | p. 355 |
| Definition of Markov operators | p. 355 |
| Irreducibility and spectrum | p. 359 |
| Continuous time Markov chains | p. 362 |
| Reversible Markov semigroups | p. 366 |
| Recurrence and transience | p. 369 |
| Spectral theory of graphs | p. 374 |
| Positive semigroups | p. 380 |
| Aspects, of positivity | p. 380 |
| Invariant subsets | p. 386 |
| Irreducibility | p. 390 |
| Renormalization | p. 293 |
| Ergodic theory | p. 395 |
| Positive semigroups on C(X) | p. 399 |
| NSA Schrodinger operators | p. 408 |
| Introduction | p. 408 |
| Bounds on the numerical range | p. 409 |
| Bounds in one space dimension | p. 412 |
| The essential spectrum of Schrodinger operators | p. 420 |
| The NSA harmonic oscillator | p. 424 |
| Semi-classical analysis | p. 427 |
| References | p. 436 |
| Index | p. 446 |
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